Enhancing Semidefinite Relaxation for Quadratically Constrained Quadratic Programming via Penalty Methods

Abstract

Quadratically constrained quadratic programming arises from a broad range of applications and is known to be among the hardest optimization problems. In recent years, semidefinite relaxation has become a popular approach for quadratically constrained quadratic programming, and many results have been reported in the literature. In this paper, we first discuss how to assess the gap between quadratically constrained quadratic programming and its semidefinite relaxation. Based on the estimated gap, we discuss how to construct an exact penalty function for quadratically constrained quadratic programming based on its semidefinite relaxation. We then introduce a special penalty method for quadratically constrained linear programming based on its semidefinite relaxation, resulting in the so-called conditionally quasi-convex relaxation. We show that the conditionally quasi-convex relaxation can provide tighter bounds than the standard semidefinite relaxation. By exploring various properties of the conditionally quasi-convex relaxation model, we develop two effective procedures, an iterative procedure and a bisection procedure, to solve the constructed conditionally quasi-convex relaxation. Promising numerical results are reported.

Appendix: Approach for Generating Disjunctive Cuts

In this appendix, we describe the approach for generating disjunctive cuts (28), which can be also found in Sections 2 and 3 of [21].

Let us define the linear operator V from ${\mathop {\mathrm{I}\mathrm{R}}\nolimits }^n$ to ${{\mathcal {S}}}^m$ by $Vx=\sum _{i=1}^nx_iV_i$, where $V_i\in {{\mathcal {S}}}^m$, $i=1,\ldots ,n$, we have for the adjoint operator $V^*$ the formula

$$\begin{aligned} V^*Z=\left( \mathrm{Tr}\displaystyle {\left(V_1Z\right)},\ldots , \mathrm{Tr}\displaystyle {\left(V_nZ\right)} \right) ^T,~~\forall Z\in {{\mathcal {S}}}^m. \end{aligned}$$

Given a polytope $P := \left\{ (x,X) \in {\mathop {\mathrm{I}\mathrm{R}}\nolimits }^n\times {{\mathcal {S}}}^n \mid Bx+ V^*X\ge b\right\}$, a point $({\hat{x}},{\hat{X}}) \in P$ and a disjunction $D: = \bigvee ^q_{k=1} (D_kx+ W_k^*X\ge d_k)$, the central issue in disjunctive programming is to demonstrate that $({\hat{x}},{\hat{X}}) \in Q$ or to find a valid inequality $\alpha ^T x +\mathrm{Tr}\displaystyle {\left(U X\right)}\ge \beta$ for Q that is violated by $({\hat{x}},{\hat{X}})$, where $Q: = \mathrm{clconv} \bigcup ^q_{k=1}\left\{ (x,X) \in P \mid D_k x + W_k^*X\ge d_k \right\}$.

The following theorem is a direct generalization of Theorem 1 in [21].

Theorem A.1

  $({\hat{x}},{\hat{X}}) \in Q$ if and only if the optimal value of the following Cut Generation Linear Program (CGLP) is nonnegative.

$$\begin{aligned}&\min \, \alpha ^T {\hat{x}} +\mathrm{Tr}\displaystyle {\left(U\hat{X}\right)}- \beta \\&\mathrm{s.~t. }\,\, B^Tu^k+D_k^Tv^k=\alpha ,~k=1,\ldots ,q,\\&\quad Vu^k+ W_kv^k=U,~k=1,\ldots ,q,\\&\quad b^Tu^k+d_k^Tv^k\ge \beta ,~k=1,\ldots ,q,\\&\quad u^k,v^k\ge 0,~k=1,\ldots ,q,\\&\quad \sum _{k=1}^q\left( \xi ^Tu^k+\xi _k^Tv^k\right) =1, \end{aligned}$$

where $\xi$ and $\xi _k$$(k = 1,\ldots , q)$ are any nonnegative vectors of conformable dimensions that satisfy $\xi _k > 0$$(k = 1, \ldots , q)$. If the optimal value of (CGLP) is negative, and $(\alpha , U,\beta , u_1, v_1, \ldots , u_q, v_q)$ is an optimal solution of (CGLP), then $\alpha ^T x +\mathrm{Tr}\displaystyle {\left(UX\right)}\ge \beta$ is a valid inequality for Q which cuts off $({\hat{x}},{\hat{X}})$.

In our computational results, the parameters $\xi$ and $\xi _k(k = 1,\ldots , q)$ in CGLP are generated in the same fashion as in [21].

Now, we describe the approach for generating disjunctive cuts in [21]. We denote by $(\hat{x}, \hat{X})$ the solution to relaxation (SDR+RLT) which we want to cut off. Let $\lambda _1 \ge \ldots \ge \lambda _q>0=\lambda _{q+1}\ldots = \lambda _n$ be eigenvalues of the matrix ${\hat{Z}}=\hat{X}-\hat{x}\hat{x}^T$, and let $p_1, \ldots , p_n$ be a corresponding set of orthonormal eigenvectors. Let $k\in \{1,\ldots ,q\}$, define

$$\begin{aligned} \eta _l^k=\min _{(x,X)\in {{\mathcal {F}}}}p^T_kx,\quad \eta _u^k=\max _{(x,X)\in {{\mathcal {F}}}}p^T_kx. \end{aligned}$$

Choose $\theta _k=p^T_k\hat{x}$. As pointed out in [21], the following disjunction can be derived by splitting the range $[\eta _l^k, \eta _u^k]$ of the function $p^T_ky$ over ${{\mathcal {F}}}$ into two intervals $[\eta _l^k, \theta _k]$ and $[\theta _k, \eta _u^k]$ and constructing a secant approximation of the function $-(p^T_ky)^2$ in each of the intervals, respectively.

$$\begin{aligned} \left[ \begin{array}{c}\eta _l^k\le p^T_kx\le \theta _k\\ \mathrm{Tr}\displaystyle {\left(p_kp_k^TX\right)}-(\eta _l^k+\theta _k)p^T_kx+\theta _k\eta _l^k\le 0\end{array}\right] \bigvee \left[ \begin{array}{c}\theta _k\le p^T_kx\le \eta _u^k \\ \mathrm{Tr}\displaystyle {\left(p_kp_k^TX\right)}-(\eta _u^k+\theta _k)p^T_kx+\theta _k\eta _u^k\le 0\end{array}\right] . \end{aligned}$$

The above disjunction can be used to derive the following disjunctive cuts by using the apparatus of CGLP:

$$\begin{aligned} \alpha _k^T x +\mathrm{Tr}\displaystyle {\left(U_kX\right)}\ge \beta _k,~~k=1,\ldots ,q. \end{aligned}$$